#318643

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2)(3,4)(5,6,10,7)(8,9,12,11)(13,14)(15,22)(16,25)(17,21)(18,20)(19,24)(23,27)(26,28);(5,7,9,12,10,6,11,8)(13,15,16,26,17,24,18,23)(14,19,25,27,21,22,20,28);(2,4)(13,15,17,24)(14,19,21,22)(16,23,18,26)(20,27,25,28);(5,8,9,7,10,12,11,6)(13,16,17,18)(14,20,21,25)(15,23,24,26)(19,27,22,28)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=i,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225839);